We also investigate the set of self-similar sets as a subset of the sub-self-similar sets and the super-self-similar sets

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Vol. LXIX, 2(2000), pp. 145–149

THE BOREL STRUCTURE OF THE COLLECTIONS OF SUB–SELF–SIMILAR SETS AND SUPER–SELF–SIMILAR SETS

M. MCCLURE and R. W. VALLIN

Abstract. We show that the sets of sub-self-similar sets and super-self-similar sets are both dense, first category,Fσ subsets ofK(R^d), the Hausdorff metric space of non-empty compact, subsets ofR^d. We also investigate the set of self-similar sets as a subset of the sub-self-similar sets and the super-self-similar sets.

1. Introduction

In [Fal1], Falconer introduced the notion of sub-self-similarity as a generalization of self-similarity and showed that sub-self-similar sets retain many of the nice properties of self-similar sets. Later in [Fal2] we find the notion of a super-self- similar set. The question arises as to how strong a generalization are these new concepts. In this paper, we quantify this question using topological notions in K(R^d), the Hausdorff metric space of non-empty compact subsets of R^d. In particular, we show that the sets of sub-self-similar sets and super-self-similar sets are both dense, first category,F_σsubsets ofK(R^d). The fact that these sets are dense could be interpreted as meaning that we have an understanding of many compact subsets ofR^d. The fact that these sets are first category indicates that most compact sets are not encompassed in these definitions. We also consider the set of self-similar sets as a subset of the sub-self-similar sets and the super-self-similar sets. In particular, we show that the sub-self-similar sets which are not self-similar are dense in the set of sub-self-similar sets, and similarly for the super-self-similar sets. This indicates that Falconer’s new concepts are a considerable generalization over the self-similar sets.

2. Definitions

We work in a fixed Euclidean space R^d. Let K(R^d) be the set of non-empty, compact subsets ofR^d. The Hausdorff metric ρonK(R^d) is defined by

ρ(A, B) = max{sup

x∈A

{dist(x, B)},sup

y∈B

{dist(y, A)}}.

Received February 23, 1999.

1980Mathematics Subject Classification(1991Revision). Primary 54B20.

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A discussion of the Hausdorff metric may be found in [Ed]Section 2.4. Of particular interest is Theorem 2.4.4, which states that K(R^d) is complete. This allows us to appeal to Baire category type arguments in K(R^d). Also of note is Ex- ercise 2.4.2, which characterizes the limit of a sequence of sets in the Hausdorff metric as follows: IfAn→A in the Hausdorff metric, then

A={x:∃{xn}^∞_n=1withxn∈An andxn →x}.

A functionT:R^d→R^d is a similarity with ratior=r(T)>0 if

|T(x)−T(y)|=r|x−y|for allx, y∈R^d.

If r <1, then T is called contractive. A fundamental result ([Ed], Thm. 4.1.3) states that ifTi:R^d→R^dis a contractive similarity for eachi∈ {1, . . . , m}, then there is a unique, nonempty, compact setE⊆R^d such that

E=∪^m_i=1Ti(E).

The setE is called self-similar.

Sub-self-similar sets are obtained by relaxing the equality to inclusion. Thus, the compact setE is sub-self-similar if there are contractive similaritiesTi: R^d →R^d fori∈ {1, . . . , m} such that

E⊆ ∪^m_i=1T_i(E).

Clearly any self-similar set is sub-self-similar. [Fal1]contains many other examples of sub-self-similar sets and describes their basic properties. The following lemma provides an example of a non-sub-self-similar set.

Lemma 1. Let E={0,1,¹₂,¹₃,¹₄, . . .}. Then E is not a sub-self-similar set.

Proof. Assume that {Ti}^m_i=1 are contractive similarities. We will show that E6⊆ ∪^m_i=1T_i(E).

Suppose first that noT_i has 0 as a fixed point. Then there is a neighborhood U of 0 such thatTi(0) ∈/ U for everyi ∈ {1, . . . , m}. Since 0 is the only cluster point ofE, it follows that U∩ ∪^m_i=1T_i(E) can contain only finitely many points.

ButU∩E is infinite, soE6⊆ ∪^m_i=1Ti(E).

Now, by reordering the set{Ti}^m_i=1if necessary, choosen≤msuch that{Ti}ⁿ_i=1 are those similarities with 0 as a fixed point. We will show thatE\ ∪ⁿ_i=1Ti(E) is infinite. Note thatT_i(E)∩E ={0} unless r(T_i) is of the specific form ^p_qⁱ

i where qi, pi ∈Nand qi ≥2. Thus if pis a prime larger thanqi for eachi∈ {1, . . . , n}, then ∪ⁿ_i=1T_i(E) will contain no number of the form _kp¹, where k ∈ N. Now the remaining portion ∪^m_i=n+1Ti(E) may contain only finitely many points of E for the reasons outlined above. Thus we again haveE6⊆ ∪^m_i=1Ti(E).

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The above argument may clearly be embedded in R^d by associating R with just one of the coordinates ofR^d. Furthermore, if E is the set in the lemma, we may obtain other non-sub-self-similar sets by scaling and translatingE. Finally, the union of such a set with any finite set will be non-sub-self-similar. Using this fact together with the fact that the finite sets are dense inK(R^d), we obtain the following important corollary.

Corollary 1. The set of non-sub-self-similar sets is dense inK(R^d).

The super-self-similar sets were introduced in [Fal2]by reversing the inclusion.

Thus, the compact set E is super-self-similar if there are contractive similarities Ti:R^d→R^d fori∈ {1, . . . , m}such that

E⊇ ∪^m_i=1Ti(E).

It again turns out that the super-self-similar sets retain some nice properties of the self-similar sets, although some additional assumption may need to be added (see [Fal2], Cor. 3.4). As with the sub-self-similar sets, we will need the fact that the set of non-super-self-similar sets is dense inK(R^d).

Lemma 2. No finite set with more than one element is super-self-similar.

Proof. Let F be a finite set with more than one element and let {T_i}^m_i=1 be contractive similarities. We will show that F 6⊇ ∪^m_i=1Ti(F). Let x be the fixed point ofT₁and lety∈F satisfy|x−y|= dist(x, F\ {x}). Then clearlyT₁(y)∈/ F

soF 6⊇ ∪^m_i=1Ti(F).

As the finite sets are dense inK(R^d), we obtain the following corollary immediately.

Corollary 2. The set of non-super-self-similar sets is dense inK(R^d).

Note that the finite sets are all sub-self-similar while the setE from Lemma 1 is super-self-similar for the set of transformations {T1(x) =¹₂x, T2(x) = ¹₃x}.

As a notational convenience, we will denote the set of self-similar sets by ss, the set of sub-self-similar sets bysssand the set of super-self-similar sets bySss.

3. The Main Results

In this section, we prove our main results. Theorem 1 states thatsssis a first category,F_σ subset ofK(R^d).

Theorem 1. The set of sub-self-similar sets may be expressed as the countable union of closed, nowhere dense subsets of K(R^d).

Proof. Form, n∈N, definesssm,nto be the set of all those sub-self-similar sets E such that there exists contractive similarities{Ti}^m_i=1 with _n¹ ≤r(Ti)≤1−_n¹,

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E ⊆ ∪^m_i=1Ti(E), and |Ti(0)| ≤ n for every i ∈ {1, . . . , m}. Clearly, ∪^∞_m=1∪^∞_n=1 sssm,nis precisely the set of sub-self-similar sets.

We first prove thatsssm,nis closed for everym, n∈N. Suppose thatEk→Ein the Hausdorff metric, whereE_k ∈sss_m,nfor everyk∈N. To eachE_kcorresponds {T_i^k}^m_i=1 such that 1/n ≤r(T_i^k)≤1−1/n, Ek ⊆ ∪^m_i=1T_i^k(Ek), and |T_i^k(0)| ≤n.

Using the standard matrix, vector representation of an affine transformation, each T_i^kmay be associated with a point,x^k_i, inR^d

2+d. The conditions on eachT_i^kensure that the set of all such points,K, is compact. By recursively choosing successively finer subsequences, we may assume that each sequence {x^k_i}^∞_k=1 is convergent to sayxi ∈K. Each point xi in turn defines a contractive similarity Ti : R^d →R^d satisfying _n¹ ≤ r(T_i) ≤ 1− ¹_n and |Ti(0)| ≤ n for every i ∈ {1, . . . , m}. The correspondence between affine transformations onR^d and points inR^d

2+d, along with the continuity of the algebraic operations, implies that T_i^k → Ti pointwise as k→ ∞. We must now show thatE ⊆ ∪^m_i=1T_i(E). Let x∈E. Then for every k∈N, there is anxk ∈Ek such that the sequence{xk}^∞_k=1 converges to x. Since Ek ⊆ ∪^m_i=1T_i^k(Ek), there is an ik ∈ {1, . . . , m} such that xk ∈ T_i^k

k(Ek). Since there are only finitely many choices forik, at least one must occur infinitely often.

Thus we have a subsequence{kj}^∞_j=1 and a fixedi∈ {1, . . . , m} such thatik_j =i for everyj. Along this subsequence we have

T_i^k^j

kj(Ek_j) =T_i^k^j(Ek_j)→Ti(E)

asj → ∞, sinceT_i^k^j →T_i pointwise and E_k_j →E in the Hausdorff metric. Thus x∈Ti(E) since xk_j →xandxk_j ∈T_i^k^j(Ek_j) for all j.

Finally, we prove thatsss_m,nis nowhere dense inK(R^d) for allm, n∈N. Since sssm,n is closed, we must simply show that it contains no open set. But this is immediate since its complement is dense inK(R^d) by Corollary 1.

The next theorem states a similar result forSss.

Theorem 2. The set of super-self-similar sets may be expressed as the countable union of closed, nowhere dense subsets ofK(R^d).

Proof. The proof of this theorem is very similar to the proof of Theorem 1. For m, n∈ N, define Sssm,n to be the set of all those super-self-similar sets E such that there exists contractive similarities {Ti}^m_i=1 with _n¹ ≤ r(Ti) ≤ 1− _n¹, E ⊇

∪^m_i=1T_i(E), and|Ti(0)| ≤nfor everyi∈ {1, . . . , m}. Using the exact construction from Theorem 1, we obtain a sequence of sets Ek →E and a sequence of transformations {T_i^k}^∞_k=1, for eachi ∈ {1, . . . , m} satisfying 1/n ≤r(T_i^k)≤ 1−1/n, Ek ⊇ ∪^m_i=1T_i^k(Ek), and|T_i^k(0)| ≤n. As before, there are transformations{Ti}^m_i=1 which are the pointwise limits as k → ∞ of {T_i^k}^∞_k=1, for each i ∈ {1, . . . , m}

and which satisfy _n¹ ≤r(Ti) ≤1− ¹_n and |Ti(0)| ≤n. We must now show that E⊇ ∪^m_i=1Ti(E). Suppose thatx∈Ti(E) for somei∈ {1, . . . , m}. Since Ek→E

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in the Hausdorff metric,Ti(Ek)→Ti(E) by the continuity ofTi. Thus for eachk we may choosexk∈Ek such thatTi(xk)→Ti(x). Thusxk→xby the continuity ofT_i⁻¹ andx∈E.

In order to show that Sss_m,n is nowhere dense in K(R^d) it again suffices to show that it contains no open set, since it is closed. But this follows immediately

from Corollary 2.

The above theorems may be somewhat improved. By allowing more general affine contractions, rather than strict similarities, we obtain the notions of sub- self-affinity and super-self-affinity. The above proofs clearly apply to the larger sets of sub-self-affine sets and super-self-affine sets.

We now turn our attention to the set of self-similar sets. It is well known thatss is dense inK(R^d). This is essentially the content of the collage theorem (see [Ba]

section 3.10, Theorem 1). This implies that sssand Sss are dense in K(R^d) as they both contain ss. In fact, ss = sss∩Sss. This implies that ss is a first category,Fσ subset ofK(R^d). Finally, we are interested in the size ofsscompared to sssand Sss. As sssand Sssare not G_δ subsets ofK(R^d), it makes no sense to consider the Baire category of their subsets (see [Ox], chapter 12). Thus we content ourselves with the following theorem which states thatssis a small subset of bothsssandSss.

Theorem 3. sss\ssis dense in sssandSss\ssis dense in Sss.

Proof. The first part is quite simple since any finite set is sub-self-similar. The finite sets are dense inK(R^d) and, therefore, dense insss\ss.

The second part is slightly more difficult. It suffices to find a class of super-self- similar sets which are not self-similar, but are dense in K(R^d). Since ssis dense in K(R^d), we show how to approximate any self-similar set with a super-self- similar set which is not self-similar. Let E be self-similar for the transformations {Ti}^m_i=1. Choose R > 0 such that Ti(BR(0)) ⊆BR(0) for each i ∈ {1, . . . , m}.

LetE1=∪^m_i=1Ti(BR(0)) and for n >1 let En =∪^m_i=1Ti(En−1). Then eachEn is super-self-similar, but not self-similar andEn →E in the Hausdorff metric.

References

[Ba] Barnsley M.,Fractals Everywhere, Academic Press, 1988.

[Ed] Edgar G. A.,Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990.

[Fal1] Falconer K. J.,Sub-Self-Similar Sets, Trans. Amer. Math. Soc.347(1995), 3121–3129.

[Fal2] ,Techniques in Fractal Geometry, John Wiley & Sons, 1997.

[Ox] Oxtoby J. C.,Measure and Category, 2nd Edition, Springer-Verlag, 1980.

M. McClure, Department of Mathematics, University of North Carolina at Asheville, Asheville, North Carolina, USA;e-mail: [email protected]

R. W. Vallin, Department of Mathematics, Slippery Rock University, Slippery Rock, Pennsylva- nia, USA;e-mail: [email protected]